R Notebook

rm(list=ls()) # clean env
options(scipen=999) # seed randomness

library(tidyr)
library(ggplot2)
library(ggExtra)
library(MASS)
library(car)
library(nnet)
library(caret)

data <- read.table('winequality-red.csv', sep=",", header=T, stringsAsFactors=T)

Summary of the data

head(data)

n <- nrow(data)  #n#
p <- ncol(data)  #p#
summary(data)

 fixed.acidity   volatile.acidity  citric.acid    residual.sugar     chlorides      
 Min.   : 4.60   Min.   :0.1200   Min.   :0.000   Min.   : 0.900   Min.   :0.01200  
 1st Qu.: 7.10   1st Qu.:0.3900   1st Qu.:0.090   1st Qu.: 1.900   1st Qu.:0.07000  
 Median : 7.90   Median :0.5200   Median :0.260   Median : 2.200   Median :0.07900  
 Mean   : 8.32   Mean   :0.5278   Mean   :0.271   Mean   : 2.539   Mean   :0.08747  
 3rd Qu.: 9.20   3rd Qu.:0.6400   3rd Qu.:0.420   3rd Qu.: 2.600   3rd Qu.:0.09000  
 Max.   :15.90   Max.   :1.5800   Max.   :1.000   Max.   :15.500   Max.   :0.61100  
 free.sulfur.dioxide total.sulfur.dioxide    density             pH       
 Min.   : 1.00       Min.   :  6.00       Min.   :0.9901   Min.   :2.740  
 1st Qu.: 7.00       1st Qu.: 22.00       1st Qu.:0.9956   1st Qu.:3.210  
 Median :14.00       Median : 38.00       Median :0.9968   Median :3.310  
 Mean   :15.87       Mean   : 46.47       Mean   :0.9967   Mean   :3.311  
 3rd Qu.:21.00       3rd Qu.: 62.00       3rd Qu.:0.9978   3rd Qu.:3.400  
 Max.   :72.00       Max.   :289.00       Max.   :1.0037   Max.   :4.010  
   sulphates         alcohol         quality     
 Min.   :0.3300   Min.   : 8.40   Min.   :3.000  
 1st Qu.:0.5500   1st Qu.: 9.50   1st Qu.:5.000  
 Median :0.6200   Median :10.20   Median :6.000  
 Mean   :0.6581   Mean   :10.42   Mean   :5.636  
 3rd Qu.:0.7300   3rd Qu.:11.10   3rd Qu.:6.000  
 Max.   :2.0000   Max.   :14.90   Max.   :8.000  

unique(data$fixed.acidity)

 [1]  7.4  7.8 11.2  7.9  7.3  7.5  6.7  5.6  8.9  8.5  8.1  7.6  6.9  6.3  7.1  8.3
[17]  5.2  5.7  8.8  6.8  4.6  7.7  8.7  6.4  6.6  8.6 10.2  7.0  7.2  9.3  8.0  9.7
[33]  6.2  5.0  4.7  8.4 10.1  9.4  9.0  8.2  6.1  5.8  9.2 11.5  5.4  9.6 12.8 11.0
[49] 11.6 12.0 15.0 10.8 11.1 10.0 12.5 11.8 10.9 10.3 11.4  9.9 10.4 13.3 10.6  9.8
[65] 13.4 10.7 11.9 12.4 12.2 13.8  9.1 13.5 10.5 12.6 14.0 13.7  9.5 12.7 12.3 15.6
[81]  5.3 11.3 13.0  6.5 12.9 14.3 15.5 11.7 13.2 15.9 12.1  5.1  4.9  5.9  6.0  5.5

unique(data$voltile.acidity)

NULL

unique(data$citric.acid)

 [1] 0.00 0.04 0.56 0.06 0.02 0.36 0.08 0.29 0.18 0.19 0.28 0.51 0.48 0.31 0.21 0.11
[17] 0.14 0.16 0.24 0.07 0.12 0.25 0.09 0.30 0.20 0.22 0.15 0.43 0.52 0.23 0.37 0.26
[33] 0.57 0.40 0.49 0.05 0.54 0.64 0.70 0.47 0.44 0.17 0.68 0.53 0.10 0.01 0.55 1.00
[49] 0.03 0.42 0.33 0.32 0.35 0.60 0.74 0.58 0.50 0.76 0.46 0.45 0.38 0.39 0.66 0.62
[65] 0.67 0.79 0.63 0.61 0.71 0.65 0.59 0.34 0.69 0.73 0.72 0.41 0.27 0.75 0.13 0.78

unique(data$residual.sugar)

 [1]  1.90  2.60  2.30  1.80  1.60  1.20  2.00  6.10  3.80  3.90  1.70  4.40  2.40
[14]  1.40  2.50 10.70  5.50  2.10  1.50  5.90  2.80  2.20  3.00  3.40  5.10  4.65
[27]  1.30  7.30  7.20  2.90  2.70  5.60  3.10  3.20  3.30  3.60  4.00  7.00  6.40
[40]  3.50 11.00  3.65  4.50  4.80  2.95  5.80  6.20  4.20  7.90  3.70  6.70  6.60
[53]  2.15  5.20  2.55 15.50  4.10  8.30  6.55  4.60  4.30  5.15  6.30  6.00  8.60
[66]  7.50  2.25  4.25  2.85  3.45  2.35  2.65  9.00  8.80  5.00  1.65  2.05  0.90
[79]  8.90  8.10  4.70  1.75  7.80 12.90 13.40  5.40 15.40  3.75 13.80  5.70 13.90

unique(data$chlorides)

  [1] 0.076 0.098 0.092 0.075 0.069 0.065 0.073 0.071 0.097 0.089 0.114 0.176 0.170
 [14] 0.368 0.086 0.341 0.077 0.082 0.106 0.084 0.085 0.080 0.105 0.083 0.103 0.066
 [27] 0.172 0.074 0.088 0.332 0.050 0.054 0.113 0.068 0.081 0.110 0.070 0.111 0.079
 [40] 0.115 0.094 0.093 0.104 0.464 0.401 0.062 0.107 0.045 0.058 0.102 0.467 0.091
 [53] 0.122 0.090 0.119 0.178 0.146 0.072 0.118 0.049 0.060 0.117 0.087 0.236 0.610
 [66] 0.095 0.100 0.360 0.067 0.270 0.099 0.046 0.061 0.056 0.039 0.059 0.101 0.057
 [79] 0.337 0.078 0.263 0.063 0.611 0.064 0.096 0.358 0.343 0.186 0.112 0.213 0.214
 [92] 0.121 0.128 0.052 0.120 0.116 0.109 0.159 0.124 0.174 0.047 0.127 0.413 0.152
[105] 0.053 0.055 0.051 0.125 0.200 0.171 0.226 0.250 0.108 0.148 0.143 0.222 0.157
[118] 0.422 0.034 0.387 0.415 0.243 0.241 0.190 0.132 0.126 0.038 0.044 0.041 0.165
[131] 0.048 0.145 0.147 0.012 0.194 0.161 0.123 0.414 0.216 0.043 0.042 0.369 0.166
[144] 0.136 0.403 0.137 0.168 0.153 0.267 0.169 0.205 0.235 0.230

unique(data$free.sulfur.dioxide)

 [1] 11.0 25.0 15.0 17.0 13.0  9.0 16.0 52.0 51.0 35.0  6.0 29.0 23.0 10.0 21.0  4.0
[17] 14.0  8.0 22.0 40.0  5.0  3.0  7.0 12.0 30.0 33.0 50.0 19.0 20.0 27.0 18.0 28.0
[33] 34.0 42.0 41.0 37.0 32.0 36.0 24.0 26.0 39.0 40.5 68.0 31.0 38.0 43.0 47.0  1.0
[49] 54.0 46.0 45.0  2.0  5.5 53.0 37.5 57.0 48.0 72.0 55.0 66.0

unique(data$total.sulfur.dioxide)

  [1]  34.0  67.0  54.0  60.0  40.0  59.0  21.0  18.0 102.0  65.0  29.0 145.0 148.0
 [14] 103.0  56.0  71.0  37.0  23.0  11.0  35.0  16.0  82.0 113.0  83.0  50.0  15.0
 [27]  30.0  19.0  87.0  46.0  14.0 114.0  12.0  96.0 119.0  73.0  45.0  10.0 110.0
 [40]  52.0 112.0  39.0  27.0  94.0  43.0  42.0  80.0  51.0  61.0 136.0  31.0 125.0
 [53]  24.0 140.0 133.0  85.0 106.0  22.0  36.0  69.0  64.0 153.0  47.0 108.0 111.0
 [66]  62.0  28.0  89.0  13.0  90.0 134.0  99.0  26.0  63.0 105.0  20.0 141.0  88.0
 [79] 129.0 128.0  86.0 121.0 101.0  44.0   8.0  49.0  38.0 143.0 144.0 127.0 126.0
 [92] 120.0  55.0  93.0  95.0  41.0  58.0  72.0  81.0 109.0  33.0  53.0  98.0  48.0
[105]  70.0  25.0 135.0  92.0  74.0  32.0  77.0 165.0  75.0 124.0  78.0 122.0  66.0
[118]  68.0  17.0  91.0  76.0 151.0 142.0 116.0 149.0  57.0 104.0  84.0 147.0 155.0
[131] 152.0   9.0 139.0 130.0   7.0 100.0 115.0   6.0  79.0 278.0 289.0 160.0  77.5
[144] 131.0

unique(data$density)

  [1] 0.99780 0.99680 0.99700 0.99800 0.99640 0.99460 0.99590 0.99430 0.99740 0.99860
 [11] 0.99690 0.99820 0.99660 0.99550 0.99620 0.99720 0.99580 0.99930 0.99570 0.99750
 [21] 0.99400 0.99760 0.99340 0.99540 0.99710 0.99560 0.99830 0.99670 0.99610 0.99840
 [31] 0.99380 0.99320 0.99650 0.99630 0.99600 0.99730 0.99880 0.99370 0.99520 0.99160
 [41] 0.99440 0.99960 0.99500 0.99810 0.99530 0.99240 0.99480 0.99695 0.99545 0.99615
 [51] 0.99940 0.99625 0.99585 0.99685 0.99655 0.99525 0.99815 0.99745 0.99270 0.99675
 [61] 0.99925 0.99565 1.00005 0.99850 0.99965 0.99575 0.99990 1.00025 0.99870 0.99935
 [71] 0.99735 0.99915 0.99910 1.00015 0.99970 1.00100 0.99790 1.00140 1.00010 0.99855
 [81] 0.99845 0.99980 0.99645 0.99865 0.99890 0.99975 0.99900 1.00150 1.00020 0.99920
 [91] 1.00080 1.00000 1.00060 1.00040 1.00180 0.99120 1.00220 1.00030 0.99490 0.99510
[101] 1.00320 0.99470 0.99950 0.99770 1.00260 1.00315 1.00210 0.99170 0.99220 0.99210
[111] 0.99788 1.00024 0.99768 0.99782 0.99761 0.99803 0.99785 0.99656 0.99488 0.99823
[121] 0.99779 0.99738 0.99701 0.99888 0.99938 0.99744 0.99668 0.99727 0.99586 0.99612
[131] 0.99676 0.99732 0.99814 0.99746 0.99708 0.99818 0.99639 0.99531 0.99786 0.99526
[141] 0.99641 0.99264 0.99682 0.99356 0.99386 0.99702 0.99693 0.99562 1.00012 0.99462
[151] 0.99939 0.99632 0.99976 0.99606 0.99154 0.99624 0.99417 0.99376 0.99832 0.99836
[161] 0.99694 0.99064 0.99672 0.99647 0.99736 0.99629 0.99689 0.99801 0.99652 0.99538
[171] 0.99594 0.99686 0.99438 0.99357 0.99628 0.99748 0.99578 0.99371 0.99522 0.99576
[181] 0.99552 0.99664 0.99614 0.99517 0.99787 0.99533 0.99536 0.99824 0.99577 0.99491
[191] 1.00289 0.99743 0.99774 0.99444 0.99892 0.99528 0.99331 0.99901 0.99674 0.99512
[201] 0.99395 0.99504 0.99516 0.99604 0.99468 0.99543 0.99791 0.99425 0.99509 0.99484
[211] 0.99834 0.99864 0.99498 0.99566 0.99408 0.99458 0.99648 0.99568 0.99613 0.99519
[221] 0.99518 0.99592 0.99654 0.99546 0.99554 0.99733 0.99669 0.99724 0.99643 0.99605
[231] 0.99658 0.99416 0.99712 0.99418 0.99596 0.99556 0.99918 0.99697 0.99378 0.99162
[241] 0.99495 0.99280 0.99603 0.99549 0.99722 0.99354 0.99635 0.99454 0.99598 0.99486
[251] 0.99007 0.99636 0.99642 0.99584 0.99506 0.99822 0.99364 0.99514 0.99854 0.99739
[261] 0.99683 0.99692 0.99756 0.99547 0.99859 0.99294 0.99634 0.99704 0.99258 0.99426
[271] 0.99747 0.99784 0.99358 0.99572 0.99769 0.99534 0.99817 0.99316 0.99471 0.99617
[281] 0.99529 0.99451 0.99479 0.99772 0.99666 0.99392 0.99388 0.99402 0.99360 0.99374
[291] 0.99523 0.99593 0.99396 0.99698 0.99020 0.99252 0.99256 0.99235 0.99352 0.99557
[301] 0.99394 0.99150 0.99379 0.99798 0.99341 0.99330 0.99684 0.99524 0.99764 0.99588
[311] 0.99473 0.99616 0.99622 0.99544 0.99728 0.99551 0.99434 0.99709 0.99384 0.99502
[321] 0.99667 0.99649 0.99716 0.99541 0.99318 0.99346 0.99599 0.99478 0.99754 0.99439
[331] 0.99633 0.99419 0.99878 0.99752 0.99428 0.99659 0.99677 0.99734 0.99678 0.99638
[341] 0.99922 0.99157 0.99718 0.99621 0.99242 0.99494 0.99729 0.99414 0.99721 0.99627
[351] 0.99569 0.99499 0.99437 0.99726 0.99456 0.99564 0.99080 0.99084 0.99350 0.99385
[361] 0.99688 0.99619 0.99476 0.99328 0.99286 0.99914 0.99521 0.99362 0.99558 0.99323
[371] 0.99191 0.99501 0.99290 0.99532 0.99796 0.99581 0.99608 0.99387 0.99448 0.99589
[381] 0.99852 0.99472 0.99587 0.99332 0.99464 0.99699 0.99725 0.99623 0.99609 0.99292
[391] 0.99420 1.00369 0.99713 0.99322 0.99706 0.99974 0.99467 0.99236 0.99705 0.99334
[401] 0.99336 1.00242 0.99182 0.99808 0.99828 0.99719 0.99542 0.99496 0.99344 0.99348
[411] 0.99459 0.99492 0.99508 0.99582 0.99555 0.99410 0.99661 0.99842 0.99489 0.99665
[421] 0.99553 0.99714 0.99631 0.99573 0.99717 0.99397 0.99646 0.99758 0.99306 0.99783
[431] 0.99765 0.99474 0.99483 0.99314 0.99574 0.99651

unique(data$pH)

 [1] 3.51 3.20 3.26 3.16 3.30 3.39 3.36 3.35 3.28 3.58 3.17 3.11 3.38 3.04 3.52 3.43
[17] 3.34 3.47 3.46 3.45 3.40 3.42 3.23 3.50 3.33 3.21 3.48 3.90 3.25 3.32 3.15 3.41
[33] 3.44 3.31 3.54 3.13 2.93 3.14 3.75 3.85 3.29 3.08 3.37 3.19 3.07 3.49 3.53 3.24
[49] 3.63 3.22 3.68 2.74 3.59 3.00 3.12 3.57 3.61 3.06 3.60 3.69 3.10 3.05 3.67 3.27
[65] 3.18 3.02 3.55 2.99 3.01 3.56 3.03 3.62 2.88 2.95 2.98 3.09 2.86 3.74 2.92 3.72
[81] 2.87 2.89 2.94 3.66 3.71 3.78 3.70 4.01 2.90

unique(data$sulphates)

 [1] 0.56 0.68 0.65 0.58 0.46 0.47 0.57 0.80 0.54 0.52 1.56 0.88 0.93 0.75 1.28 0.50
[17] 1.08 0.53 0.91 0.63 0.59 0.55 0.66 0.60 0.73 0.48 0.83 0.51 0.90 1.20 0.74 0.64
[33] 0.77 0.71 0.62 0.39 0.79 0.95 0.82 1.12 1.14 0.78 1.95 1.22 1.98 0.61 1.31 0.69
[49] 0.67 0.70 0.49 0.92 2.00 0.72 1.59 0.33 1.02 0.97 0.85 0.43 1.03 0.86 0.76 1.61
[65] 1.09 0.84 0.96 0.45 1.26 0.87 0.81 1.00 1.36 1.18 0.89 0.98 1.13 1.04 1.11 0.99
[81] 1.07 0.44 1.06 1.05 0.42 1.17 1.62 0.94 1.34 1.16 1.10 0.40 1.15 0.37 1.33 1.01

unique(data$alcohol)

 [1]  9.400000  9.800000 10.000000  9.500000 10.500000  9.200000  9.900000  9.100000
 [9]  9.300000  9.000000  9.700000 10.100000 10.600000  9.600000 10.800000 10.300000
[17] 13.100000 10.200000 10.900000 10.700000 12.900000 10.400000 13.000000 14.000000
[25] 11.500000 11.400000 12.400000 11.000000 12.200000 12.800000 12.600000 12.500000
[33] 11.700000 11.300000 12.300000 12.000000 11.900000 11.800000  8.700000 13.300000
[41] 11.200000 11.600000 11.100000 13.400000 12.100000  8.400000 12.700000 14.900000
[49] 13.200000 13.600000 13.500000 10.033333  9.550000  8.500000 11.066667  9.566667
[57] 10.550000  8.800000 13.566667 11.950000  9.950000  9.233333  9.250000  9.050000
[65] 10.750000

unique(data$quality)

[1] 5 6 7 4 8 3

data$label <- with(data, ifelse(quality >= 7, 'great', 
                         ifelse(quality >= 5, 'good', 'poor')))
data$y <- with(data, ifelse(quality >= 7, 1, 0))

df <- data[,1:11]
cat <- data[,13]
lab <- data[,14]

pairs(df)

colMeans(data[,1:12])

       fixed.acidity     volatile.acidity          citric.acid       residual.sugar 
          8.31963727           0.52782051           0.27097561           2.53880550 
           chlorides  free.sulfur.dioxide total.sulfur.dioxide              density 
          0.08746654          15.87492183          46.46779237           0.99674668 
                  pH            sulphates              alcohol              quality 
          3.31111320           0.65814884          10.42298311           5.63602251 

mvec <- colMeans(df)   # sample mean vector
covM <- cov(df)          # sample covariance matrix
corM <- cor(df)        # sample correlation matrix
det(cov(df))           # generalized sample variance

[1] 0.00000000003478418

sum(diag(cov(df)))     # total sample variance 

[1] 1197.797

Assessing univariate normality

FindcrikChi <- function(n, p, alpha=0.5, N=1000){
    
    cricvec <- rep(0, N)  #vector for the rQ result collection#
    
    for(i in 1:N){
        #iteration to estimate rQ#
        numvec <- rchisq(n, p)  #generate a data set of size n, degree of freedom=p#
        d <- sort(numvec)
        q <- qchisq((1:n-0.5)/n, p)
        cricvec[i] <- cor(d,q)      
    }
    
    scricvec <- sort(cricvec)
    cN <- ceiling(N* alpha) #to be on the safe side I use ceiling instead of floor(), take the 'worst' alpha*N cor as rQ, everything lower than that is deemed as rejection#
    cricvalue <- scricvec[cN]
    result <- list(cN, cricvalue, scricvec)
    return(result)
}

critic <- FindcrikChi(n, p-1)
critic[[2]]

[1] 0.9993465

DensityPlots <- function(data_set){

  for (col in names(data_set)){
    print(mean(data_set[[col]]))
    qqc <- qqnorm(data_set[[col]], main = paste("QQ - Plot: ", col))
    corqq <- cor(qqc$x, qqc$y) 
  
    if (round(corqq,2) >= round(critic[[2]],3)){
      qqline(data_set[[col]], col='blue', lwd=2)
      print(paste('Data ', col, ' is Normally Distributed! with: ', round(corqq,3)))
    } else {
      qqline(data_set[[col]], col='orange', lwd=2)
      print(paste('Data ', col, ' is NOT Normally Distributed! with: ', round(corqq,3)))
    }
  
    for ( i in 1:ncol(data_set)){
      if (col != names(data_set[i])){
        j <- names(data_set[i])
        df_mean <- as.data.frame(colMeans(data_set[c(col,  j)]))
      
        plot <- ggplot(data = data_set) + 
          geom_point(mapping = aes(x = .data[[col]], y = .data[[j]])) +
          geom_point(data=t(df_mean),  mapping=aes(x = .data[[col]], y = .data[[j]]), col="red")
        print(ggMarginal(plot, type="densigram"))
      
      # or standard R
      # plot(data_set[[col]], data_set[,i], col='blue', lwd=2, xlab=col, ylab=j)
      # points(mean(data_set[[col]]), mean(data_set[,i]), col='red', lwd=8)
      
      print(paste(col, ' vs ', names(data_set[i]), ': ', 
                  cov(data_set[[col]], data_set[,i])))

    }
  }
  }
}

DensityPlots(df)

[1] 8.319637
[1] "Data  fixed.acidity  is NOT Normally Distributed! with:  0.971"
[1] "fixed.acidity  vs  volatile.acidity :  -0.0798514168351465"
[1] "fixed.acidity  vs  citric.acid :  0.227820003663115"
[1] "fixed.acidity  vs  residual.sugar :  0.281756262322901"
[1] "fixed.acidity  vs  chlorides :  0.0076786924869345"
[1] "fixed.acidity  vs  free.sulfur.dioxide :  -2.8009214927039"
[1] "fixed.acidity  vs  total.sulfur.dioxide :  -6.48234585758778"
[1] "fixed.acidity  vs  density :  0.00219522357567034"
[1] "fixed.acidity  vs  pH :  -0.183585703596037"
[1] "fixed.acidity  vs  sulphates :  0.0540100915700598"
[1] "fixed.acidity  vs  alcohol :  -0.114421153396092"
[1] 0.5278205

[1] "Data  volatile.acidity  is NOT Normally Distributed! with:  0.987"
[1] "volatile.acidity  vs  fixed.acidity :  -0.0798514168351465"
[1] "volatile.acidity  vs  citric.acid :  -0.01927162077597"
[1] "volatile.acidity  vs  residual.sugar :  0.000484190975899359"
[1] "volatile.acidity  vs  chlorides :  0.00051658691954687"
[1] "volatile.acidity  vs  free.sulfur.dioxide :  -0.0196735903854177"
[1] "volatile.acidity  vs  total.sulfur.dioxide :  0.450425692371875"
[1] "volatile.acidity  vs  density :  0.00000744366515837123"
[1] "volatile.acidity  vs  pH :  0.0064946993036167"
[1] "volatile.acidity  vs  sulphates :  -0.00792143384358653"
[1] "volatile.acidity  vs  alcohol :  -0.0386002214306344"
[1] 0.2709756

[1] "Data  citric.acid  is NOT Normally Distributed! with:  0.977"
[1] "citric.acid  vs  fixed.acidity :  0.227820003663115"
[1] "citric.acid  vs  volatile.acidity :  -0.01927162077597"
[1] "citric.acid  vs  residual.sugar :  0.0394342699716109"
[1] "citric.acid  vs  chlorides :  0.00186872477792362"
[1] "citric.acid  vs  free.sulfur.dioxide :  -0.124252113922891"
[1] "citric.acid  vs  total.sulfur.dioxide :  0.227697274031564"
[1] "citric.acid  vs  density :  0.000134174581031167"
[1] "citric.acid  vs  pH :  -0.0162975823437834"
[1] "citric.acid  vs  sulphates :  0.0103277145212003"
[1] "citric.acid  vs  alcohol :  0.0228151729295766"
[1] 2.538806

[1] "Data  residual.sugar  is NOT Normally Distributed! with:  0.752"
[1] "residual.sugar  vs  fixed.acidity :  0.281756262322901"
[1] "residual.sugar  vs  volatile.acidity :  0.000484190975899359"
[1] "residual.sugar  vs  citric.acid :  0.0394342699716109"
[1] "residual.sugar  vs  chlorides :  0.00369017590390114"
[1] "residual.sugar  vs  free.sulfur.dioxide :  2.75861145224526"
[1] "residual.sugar  vs  total.sulfur.dioxide :  9.4164414789907"
[1] "residual.sugar  vs  density :  0.000945410861841846"
[1] "residual.sugar  vs  pH :  -0.0186442889838064"
[1] "residual.sugar  vs  sulphates :  0.00132094135806093"
[1] "residual.sugar  vs  alcohol :  0.0632189597926113"
[1] 0.08746654

[1] "Data  chlorides  is NOT Normally Distributed! with:  0.695"
[1] "chlorides  vs  fixed.acidity :  0.0076786924869345"
[1] "chlorides  vs  volatile.acidity :  0.00051658691954687"
[1] "chlorides  vs  citric.acid :  0.00186872477792362"
[1] "chlorides  vs  residual.sugar :  0.00369017590390114"
[1] "chlorides  vs  free.sulfur.dioxide :  0.00273830307740836"
[1] "chlorides  vs  total.sulfur.dioxide :  0.0733867502451861"
[1] "chlorides  vs  density :  0.000017821756780873"
[1] "chlorides  vs  pH :  -0.00192574495871559"
[1] "chlorides  vs  sulphates :  0.00296187794937543"
[1] "chlorides  vs  alcohol :  -0.0110915177743286"
[1] 15.87492

[1] "Data  free.sulfur.dioxide  is NOT Normally Distributed! with:  0.95"
[1] "free.sulfur.dioxide  vs  fixed.acidity :  -2.8009214927039"
[1] "free.sulfur.dioxide  vs  volatile.acidity :  -0.0196735903854177"
[1] "free.sulfur.dioxide  vs  citric.acid :  -0.124252113922891"
[1] "free.sulfur.dioxide  vs  residual.sugar :  2.75861145224526"
[1] "free.sulfur.dioxide  vs  chlorides :  0.00273830307740836"
[1] "free.sulfur.dioxide  vs  total.sulfur.dioxide :  229.737520947463"
[1] "free.sulfur.dioxide  vs  density :  -0.000433250416209755"
[1] "free.sulfur.dioxide  vs  pH :  0.113653090831958"
[1] "free.sulfur.dioxide  vs  sulphates :  0.0915924709670703"
[1] "free.sulfur.dioxide  vs  alcohol :  -0.773698400361301"
[1] 46.46779

[1] "Data  total.sulfur.dioxide  is NOT Normally Distributed! with:  0.934"
[1] "total.sulfur.dioxide  vs  fixed.acidity :  -6.48234585758778"
[1] "total.sulfur.dioxide  vs  volatile.acidity :  0.450425692371875"
[1] "total.sulfur.dioxide  vs  citric.acid :  0.227697274031564"
[1] "total.sulfur.dioxide  vs  residual.sugar :  9.4164414789907"
[1] "total.sulfur.dioxide  vs  chlorides :  0.0733867502451861"
[1] "total.sulfur.dioxide  vs  free.sulfur.dioxide :  229.737520947463"
[1] "total.sulfur.dioxide  vs  density :  0.00442472714485978"
[1] "total.sulfur.dioxide  vs  pH :  -0.337698792502511"
[1] "total.sulfur.dioxide  vs  sulphates :  0.239471004640729"
[1] "total.sulfur.dioxide  vs  alcohol :  -7.20929789503922"
[1] 0.9967467

[1] "Data  density  is Normally Distributed! with:  0.995"
[1] "density  vs  fixed.acidity :  0.00219522357567034"
[1] "density  vs  volatile.acidity :  0.00000744366515837123"
[1] "density  vs  citric.acid :  0.000134174581031167"
[1] "density  vs  residual.sugar :  0.000945410861841846"
[1] "density  vs  chlorides :  0.000017821756780873"
[1] "density  vs  free.sulfur.dioxide :  -0.000433250416209755"
[1] "density  vs  total.sulfur.dioxide :  0.00442472714485978"
[1] "density  vs  pH :  -0.0000995639480166344"
[1] "density  vs  sulphates :  0.0000475096184959153"
[1] "density  vs  alcohol :  -0.000997951789525837"
[1] 3.311113

[1] "Data  pH  is Normally Distributed! with:  0.997"
[1] "pH  vs  fixed.acidity :  -0.183585703596037"
[1] "pH  vs  volatile.acidity :  0.0064946993036167"
[1] "pH  vs  citric.acid :  -0.0162975823437834"
[1] "pH  vs  residual.sugar :  -0.0186442889838064"
[1] "pH  vs  chlorides :  -0.00192574495871559"
[1] "pH  vs  free.sulfur.dioxide :  0.113653090831958"
[1] "pH  vs  total.sulfur.dioxide :  -0.337698792502511"
[1] "pH  vs  density :  -0.0000995639480166344"
[1] "pH  vs  sulphates :  -0.0051461858201426"
[1] "pH  vs  alcohol :  0.0338316166393107"
[1] 0.6581488

[1] "Data  sulphates  is NOT Normally Distributed! with:  0.912"
[1] "sulphates  vs  fixed.acidity :  0.0540100915700598"
[1] "sulphates  vs  volatile.acidity :  -0.00792143384358653"
[1] "sulphates  vs  citric.acid :  0.0103277145212003"
[1] "sulphates  vs  residual.sugar :  0.00132094135806093"
[1] "sulphates  vs  chlorides :  0.00296187794937543"
[1] "sulphates  vs  free.sulfur.dioxide :  0.0915924709670703"
[1] "sulphates  vs  total.sulfur.dioxide :  0.239471004640729"
[1] "sulphates  vs  density :  0.0000475096184959153"
[1] "sulphates  vs  pH :  -0.0051461858201426"
[1] "sulphates  vs  alcohol :  0.0169067772332677"
[1] 10.42298

[1] "Data  alcohol  is NOT Normally Distributed! with:  0.964"
[1] "alcohol  vs  fixed.acidity :  -0.114421153396092"
[1] "alcohol  vs  volatile.acidity :  -0.0386002214306344"
[1] "alcohol  vs  citric.acid :  0.0228151729295766"
[1] "alcohol  vs  residual.sugar :  0.0632189597926113"
[1] "alcohol  vs  chlorides :  -0.0110915177743286"
[1] "alcohol  vs  free.sulfur.dioxide :  -0.773698400361301"
[1] "alcohol  vs  total.sulfur.dioxide :  -7.20929789503922"
[1] "alcohol  vs  density :  -0.000997951789525837"
[1] "alcohol  vs  pH :  0.0338316166393107"
[1] "alcohol  vs  sulphates :  0.0169067772332677"

Transformation

df_normal <- data.frame(matrix(nrow=n, ncol = 11))
colnames(df_normal) <- names(df)

normal <- c()
not_normal <- c()
for (col in names(df)){
  tryCatch(
        {
        boxcoxTransc <- boxcox(df[[col]] ~ 1,lambda=seq(-2.5, 2.5,.01))
        title(col)
  
        flagidx <- which(boxcoxTransc$y==max(boxcoxTransc$y))
        optlam <- boxcoxTransc$x[flagidx]
        vec <- df[[col]]

        transvec <- (vec^optlam-1)/optlam  #according to (4-34)#

        # transformed data#
        qqts <- qqnorm(transvec, main = paste("QQ - Plot: ", col))
        cortrans <- cor(qqts$x, qqts$y)
        
        },
        error = function(cond) {
            message(paste("Data NOT transformed: ", col))
            message("Here's the original error message:")
            message(conditionMessage(cond))
            # Choose a return value in case of error
            qqts <- qqnorm(df[[col]], main = paste("QQ - Plot: ", col))
            qqline(df[[col]], col='orange', lwd=2)
            cortrans <- cor(qqts$x, qqts$y)
            return(cortrans)
        },
        finally = {
          if (round(cortrans, 2) >= round(critic[[2]], 3)){
          normal <- append(normal, col)
          qqline(transvec, col='blue', lwd=2)
          print(paste('Data ', col, ' is Normally Distributed! with: ', round(cortrans,3)))
          df_normal[[col]] <- transvec
        } else {
          not_normal <- append(not_normal, col)
          qqline(transvec, col='orange', lwd=2)
          print(paste('Data ', col, ' is NOT Normally Distributed! with: ',
                      round(cortrans,3)))
        } 
      }
    )
}

[1] "Data  fixed.acidity  is Normally Distributed! with:  0.997"

[1] "Data  volatile.acidity  is Normally Distributed! with:  0.998"

Data NOT transformed:  citric.acid
Here's the original error message:
response variable must be positive

[1] "Data  citric.acid  is Normally Distributed! with:  0.998"

[1] "Data  residual.sugar  is NOT Normally Distributed! with:  0.988"

[1] "Data  chlorides  is NOT Normally Distributed! with:  0.933"

[1] "Data  free.sulfur.dioxide  is NOT Normally Distributed! with:  0.994"

[1] "Data  total.sulfur.dioxide  is Normally Distributed! with:  0.995"

[1] "Data  density  is Normally Distributed! with:  0.996"

[1] "Data  pH  is Normally Distributed! with:  0.998"

[1] "Data  sulphates  is Normally Distributed! with:  0.998"

[1] "Data  alcohol  is NOT Normally Distributed! with:  0.985"

unique(df$citric.acid)

 [1] 0.00 0.04 0.56 0.06 0.02 0.36 0.08 0.29 0.18 0.19 0.28 0.51 0.48 0.31 0.21 0.11
[17] 0.14 0.16 0.24 0.07 0.12 0.25 0.09 0.30 0.20 0.22 0.15 0.43 0.52 0.23 0.37 0.26
[33] 0.57 0.40 0.49 0.05 0.54 0.64 0.70 0.47 0.44 0.17 0.68 0.53 0.10 0.01 0.55 1.00
[49] 0.03 0.42 0.33 0.32 0.35 0.60 0.74 0.58 0.50 0.76 0.46 0.45 0.38 0.39 0.66 0.62
[65] 0.67 0.79 0.63 0.61 0.71 0.65 0.59 0.34 0.69 0.73 0.72 0.41 0.27 0.75 0.13 0.78

df_norm <- df_normal[normal]
df_norm <- within(df_norm,  rm('citric.acid'))
pairs(df_norm)

DensityPlots(df_norm)

[1] 1.123393
[1] "Data  fixed.acidity  is Normally Distributed! with:  0.997"
[1] "fixed.acidity  vs  volatile.acidity :  -0.00325578336388052"
[1] "fixed.acidity  vs  total.sulfur.dioxide :  -0.00445079118867292"
[1] "fixed.acidity  vs  density :  0.0000620357104893582"
[1] "fixed.acidity  vs  pH :  -0.00154673826556414"
[1] "fixed.acidity  vs  sulphates :  0.00316375032430066"
[1] -0.5907973

[1] "Data  volatile.acidity  is Normally Distributed! with:  0.998"
[1] "volatile.acidity  vs  fixed.acidity :  -0.00325578336388052"
[1] "volatile.acidity  vs  total.sulfur.dioxide :  0.0201015425654002"
[1] "volatile.acidity  vs  density :  0.0000183855215118722"
[1] "volatile.acidity  vs  pH :  0.00269943316805013"
[1] "volatile.acidity  vs  sulphates :  -0.0282823793958877"
[1] 3.960085

[1] "Data  total.sulfur.dioxide  is Normally Distributed! with:  0.995"
[1] "total.sulfur.dioxide  vs  fixed.acidity :  -0.00445079118867292"
[1] "total.sulfur.dioxide  vs  volatile.acidity :  0.0201015425654002"
[1] "total.sulfur.dioxide  vs  density :  0.00016722798065971"
[1] "total.sulfur.dioxide  vs  pH :  -0.000706287076722766"
[1] "total.sulfur.dioxide  vs  sulphates :  0.011014698090896"
[1] -0.003278254

[1] "Data  density  is Normally Distributed! with:  0.996"
[1] "density  vs  fixed.acidity :  0.0000620357104893582"
[1] "density  vs  volatile.acidity :  0.0000183855215118722"
[1] "density  vs  total.sulfur.dioxide :  0.00016722798065971"
[1] "density  vs  pH :  -0.0000292094469761482"
[1] "density  vs  sulphates :  0.000115258678596354"
[1] 1.17496

[1] "Data  pH  is Normally Distributed! with:  0.998"
[1] "pH  vs  fixed.acidity :  -0.00154673826556414"
[1] "pH  vs  volatile.acidity :  0.00269943316805013"
[1] "pH  vs  total.sulfur.dioxide :  -0.000706287076722766"
[1] "pH  vs  density :  -0.0000292094469761482"
[1] "pH  vs  sulphates :  -0.0017818484264631"
[1] -0.6092693

[1] "Data  sulphates  is Normally Distributed! with:  0.998"
[1] "sulphates  vs  fixed.acidity :  0.00316375032430066"
[1] "sulphates  vs  volatile.acidity :  -0.0282823793958877"
[1] "sulphates  vs  total.sulfur.dioxide :  0.011014698090896"
[1] "sulphates  vs  density :  0.000115258678596354"
[1] "sulphates  vs  pH :  -0.0017818484264631"

df_to_scale <- df[not_normal]
df_to_scale$citric.acid <- df$citric.acid

scale_data <- as.data.frame(scale(df_to_scale))
pairs(scale_data)

DensityPlots(scale_data)

[1] -0.0000000000000001156003
[1] "Data  residual.sugar  is NOT Normally Distributed! with:  0.752"
[1] "residual.sugar  vs  chlorides :  0.0556095352035322"
[1] "residual.sugar  vs  free.sulfur.dioxide :  0.187048995104287"
[1] "residual.sugar  vs  alcohol :  0.0420754372097311"
[1] "residual.sugar  vs  citric.acid :  0.143577161570314"
[1] 0.00000000000000008613634

[1] "Data  chlorides  is NOT Normally Distributed! with:  0.695"
[1] "chlorides  vs  residual.sugar :  0.0556095352035322"
[1] "chlorides  vs  free.sulfur.dioxide :  0.00556214700478112"
[1] "chlorides  vs  alcohol :  -0.221140544788283"
[1] "chlorides  vs  citric.acid :  0.203822913829042"
[1] -0.00000000000000005600528

[1] "Data  free.sulfur.dioxide  is NOT Normally Distributed! with:  0.95"
[1] "free.sulfur.dioxide  vs  residual.sugar :  0.187048995104287"
[1] "free.sulfur.dioxide  vs  chlorides :  0.00556214700478112"
[1] "free.sulfur.dioxide  vs  alcohol :  -0.0694083535649999"
[1] "free.sulfur.dioxide  vs  citric.acid :  -0.0609781291923049"
[1] 0.00000000000000008786086

[1] "Data  alcohol  is NOT Normally Distributed! with:  0.964"
[1] "alcohol  vs  residual.sugar :  0.0420754372097311"
[1] "alcohol  vs  chlorides :  -0.221140544788283"
[1] "alcohol  vs  free.sulfur.dioxide :  -0.0694083535649999"
[1] "alcohol  vs  citric.acid :  0.109903246641567"
[1] -0.00000000000000009207575

[1] "Data  citric.acid  is NOT Normally Distributed! with:  0.977"
[1] "citric.acid  vs  residual.sugar :  0.143577161570314"
[1] "citric.acid  vs  chlorides :  0.203822913829042"
[1] "citric.acid  vs  free.sulfur.dioxide :  -0.0609781291923049"
[1] "citric.acid  vs  alcohol :  0.109903246641567"

log_scale <- log(df_to_scale)
pairs(log_scale)

DensityPlots(log_scale)

[1] 0.8502318
[1] "Data  residual.sugar  is NOT Normally Distributed! with:  0.925"
[1] "residual.sugar  vs  chlorides :  0.0136963986844303"
[1] "residual.sugar  vs  free.sulfur.dioxide :  0.0229861717575993"
[1] "residual.sugar  vs  alcohol :  0.00281540290773283"
[1] "residual.sugar  vs  citric.acid :  NaN"
[1] -2.505462

[1] "Data  chlorides  is NOT Normally Distributed! with:  0.91"
[1] "chlorides  vs  residual.sugar :  0.0136963986844303"
[1] "chlorides  vs  free.sulfur.dioxide :  -0.00304888609384212"
[1] "chlorides  vs  alcohol :  -0.00989314903575179"
[1] "chlorides  vs  citric.acid :  NaN"
[1] 2.546132

[1] "Data  free.sulfur.dioxide  is NOT Normally Distributed! with:  0.992"
[1] "free.sulfur.dioxide  vs  residual.sugar :  0.0229861717575993"
[1] "free.sulfur.dioxide  vs  chlorides :  -0.00304888609384212"
[1] "free.sulfur.dioxide  vs  alcohol :  -0.00570807621737924"
[1] "free.sulfur.dioxide  vs  citric.acid :  NaN"
[1] 2.339021

[1] "Data  alcohol  is NOT Normally Distributed! with:  0.973"
[1] "alcohol  vs  residual.sugar :  0.00281540290773283"
[1] "alcohol  vs  chlorides :  -0.00989314903575179"
[1] "alcohol  vs  free.sulfur.dioxide :  -0.00570807621737924"
[1] "alcohol  vs  citric.acid :  NaN"
[1] -Inf

Error in plot.window(...) : need finite 'ylim' values

process <- preProcess(df_to_scale, method=c("range"))
norm_scale <- predict(process, df_to_scale)
pairs(norm_scale)

DensityPlots(norm_scale)

[1] 0.112247
[1] "Data  residual.sugar  is NOT Normally Distributed! with:  0.752"
[1] "residual.sugar  vs  chlorides :  0.000421956217428721"
[1] "residual.sugar  vs  free.sulfur.dioxide :  0.00266121112506778"
[1] "residual.sugar  vs  alcohol :  0.000666163959879993"
[1] "residual.sugar  vs  citric.acid :  0.00270097739531581"
[1] 0.1259875

[1] "Data  chlorides  is NOT Normally Distributed! with:  0.695"
[1] "chlorides  vs  residual.sugar :  0.000421956217428721"
[1] "chlorides  vs  free.sulfur.dioxide :  0.0000643867261729257"
[1] "chlorides  vs  alcohol :  -0.00284872679448532"
[1] "chlorides  vs  citric.acid :  0.00311974086464712"
[1] 0.2095059

[1] "Data  free.sulfur.dioxide  is NOT Normally Distributed! with:  0.95"
[1] "free.sulfur.dioxide  vs  residual.sugar :  0.00266121112506778"
[1] "free.sulfur.dioxide  vs  chlorides :  0.0000643867261729257"
[1] "free.sulfur.dioxide  vs  alcohol :  -0.00167648624130293"
[1] "free.sulfur.dioxide  vs  citric.acid :  -0.00175002977356185"
[1] 0.3112282

[1] "Data  alcohol  is NOT Normally Distributed! with:  0.964"
[1] "alcohol  vs  residual.sugar :  0.000666163959879993"
[1] "alcohol  vs  chlorides :  -0.00284872679448532"
[1] "alcohol  vs  free.sulfur.dioxide :  -0.00167648624130293"
[1] "alcohol  vs  citric.acid :  0.00351002660455024"
[1] 0.2709756

[1] "Data  citric.acid  is NOT Normally Distributed! with:  0.977"
[1] "citric.acid  vs  residual.sugar :  0.00270097739531581"
[1] "citric.acid  vs  chlorides :  0.00311974086464712"
[1] "citric.acid  vs  free.sulfur.dioxide :  -0.00175002977356185"
[1] "citric.acid  vs  alcohol :  0.00351002660455024"

df_stndardized <- df_to_scale
for (col in names(df_to_scale)){
  df_stndardized[[col]] <- (df_to_scale[[col]] - mean(df_to_scale[[col]])) / sd(df_to_scale[[col]])
}

pairs(df_stndardized)

DensityPlots(df_stndardized)

[1] -0.0000000000000001156003
[1] "Data  residual.sugar  is NOT Normally Distributed! with:  0.752"
[1] "residual.sugar  vs  chlorides :  0.0556095352035322"
[1] "residual.sugar  vs  free.sulfur.dioxide :  0.187048995104287"
[1] "residual.sugar  vs  alcohol :  0.0420754372097311"
[1] "residual.sugar  vs  citric.acid :  0.143577161570314"
[1] 0.00000000000000008888973

[1] "Data  chlorides  is NOT Normally Distributed! with:  0.695"
[1] "chlorides  vs  residual.sugar :  0.0556095352035322"
[1] "chlorides  vs  free.sulfur.dioxide :  0.00556214700478112"
[1] "chlorides  vs  alcohol :  -0.221140544788283"
[1] "chlorides  vs  citric.acid :  0.203822913829042"
[1] -0.00000000000000005600528

[1] "Data  free.sulfur.dioxide  is NOT Normally Distributed! with:  0.95"
[1] "free.sulfur.dioxide  vs  residual.sugar :  0.187048995104287"
[1] "free.sulfur.dioxide  vs  chlorides :  0.00556214700478112"
[1] "free.sulfur.dioxide  vs  alcohol :  -0.0694083535649999"
[1] "free.sulfur.dioxide  vs  citric.acid :  -0.0609781291923049"
[1] 0.00000000000000008080805

[1] "Data  alcohol  is NOT Normally Distributed! with:  0.964"
[1] "alcohol  vs  residual.sugar :  0.0420754372097311"
[1] "alcohol  vs  chlorides :  -0.221140544788283"
[1] "alcohol  vs  free.sulfur.dioxide :  -0.0694083535649999"
[1] "alcohol  vs  citric.acid :  0.109903246641567"
[1] -0.00000000000000009207575

[1] "Data  citric.acid  is NOT Normally Distributed! with:  0.977"
[1] "citric.acid  vs  residual.sugar :  0.143577161570314"
[1] "citric.acid  vs  chlorides :  0.203822913829042"
[1] "citric.acid  vs  free.sulfur.dioxide :  -0.0609781291923049"
[1] "citric.acid  vs  alcohol :  0.109903246641567"

chiforbi <- FindcrikChi(n, 2)

BivariateQQ <- function(data_set){
  for (col in names(data_set)){
    for ( i in 1:ncol(data_set)){
      if (col != names(data_set[i])){
        dat <- c(col, names(data_set[i]))
        X <- (data_set[dat])
        mu <- colMeans(data_set[dat])
        S <- cov(data_set[dat])
        
        result <- c() 
        tryCatch(
        {
          for (row in 1:nrow(X)){
            v <- as.matrix(X[row,])
            result[row] <- mahalanobis(v, mu, S)
            #result[row] <- (v-mu)%*%solve(S)%*%t(v-mu)
            y <- sort(result)
            # The second parameter is now '2' 
            # because we have only two variables (bivariate)
            x <- qchisq(1:length(result)/(length(result)+1), 2)
          }
          plot(x, y) 
          if (round(cor(x, y),2) >=  round(chiforbi[[2]],3)){
            abline(0,1, col='blue', lwd=2)
            print(paste(col, ' vs ', names(data_set[i]), 
                      ' is Normally Distributed! with: ', round(cor(x, y),3)))
          } else {
            abline(0,1, col='orange', lwd=2)
            print(paste(col, ' vs ', names(data_set[i]), 
                      ' is NOT Normally Distributed! with: ', round(cor(x, y),3)))
          }
        },
        error = function(cond) {
            message(paste("Data NOT calculated: ", col, names(data_set[i])))
            message("Here's the original error message:")
            message(conditionMessage(cond))
            # Choose a return value in case of error
            NA
        })
      }    
    }
  }
}

round(chiforbi[[2]],3)

[1] 0.999

BivariateQQ(df_norm)

[1] "fixed.acidity  vs  volatile.acidity  is Normally Distributed! with:  0.996"

[1] "fixed.acidity  vs  total.sulfur.dioxide  is Normally Distributed! with:  0.996"

[1] "fixed.acidity  vs  density  is NOT Normally Distributed! with:  0.989"

[1] "fixed.acidity  vs  pH  is NOT Normally Distributed! with:  0.985"

[1] "fixed.acidity  vs  sulphates  is Normally Distributed! with:  0.996"

[1] "volatile.acidity  vs  fixed.acidity  is Normally Distributed! with:  0.996"

[1] "volatile.acidity  vs  total.sulfur.dioxide  is NOT Normally Distributed! with:  0.99"

[1] "volatile.acidity  vs  density  is NOT Normally Distributed! with:  0.993"

[1] "volatile.acidity  vs  pH  is NOT Normally Distributed! with:  0.991"

[1] "volatile.acidity  vs  sulphates  is NOT Normally Distributed! with:  0.992"

[1] "total.sulfur.dioxide  vs  fixed.acidity  is Normally Distributed! with:  0.996"

[1] "total.sulfur.dioxide  vs  volatile.acidity  is NOT Normally Distributed! with:  0.99"

[1] "total.sulfur.dioxide  vs  density  is NOT Normally Distributed! with:  0.988"

[1] "total.sulfur.dioxide  vs  pH  is NOT Normally Distributed! with:  0.992"

[1] "total.sulfur.dioxide  vs  sulphates  is NOT Normally Distributed! with:  0.988"

[1] "density  vs  fixed.acidity  is NOT Normally Distributed! with:  0.989"

[1] "density  vs  volatile.acidity  is NOT Normally Distributed! with:  0.993"

[1] "density  vs  total.sulfur.dioxide  is NOT Normally Distributed! with:  0.988"

[1] "density  vs  pH  is NOT Normally Distributed! with:  0.968"

[1] "density  vs  sulphates  is NOT Normally Distributed! with:  0.992"

[1] "pH  vs  fixed.acidity  is NOT Normally Distributed! with:  0.985"

[1] "pH  vs  volatile.acidity  is NOT Normally Distributed! with:  0.991"

[1] "pH  vs  total.sulfur.dioxide  is NOT Normally Distributed! with:  0.992"

[1] "pH  vs  density  is NOT Normally Distributed! with:  0.968"

[1] "pH  vs  sulphates  is NOT Normally Distributed! with:  0.985"

[1] "sulphates  vs  fixed.acidity  is Normally Distributed! with:  0.996"

[1] "sulphates  vs  volatile.acidity  is NOT Normally Distributed! with:  0.992"

[1] "sulphates  vs  total.sulfur.dioxide  is NOT Normally Distributed! with:  0.988"

[1] "sulphates  vs  density  is NOT Normally Distributed! with:  0.992"

[1] "sulphates  vs  pH  is NOT Normally Distributed! with:  0.985"

Data is NO normally distributed

Homogenity Check

df2 <- cbind(df_norm, df_to_scale, data[, 13:14])

df_grate <- df2[df2$label=='great', ]
df_good <- df2[df2$label=='good', ]
df_poor <- df2[df2$label=='poor', ]
df_1 <- df2[df2$y== 1, ]
df_0 <- df2[df2$y ==0, ]

Trying to calculate Box’s M Test manually

n1 <- nrow(df_grate)
n2 <- nrow(df_good)
n3 <- nrow(df_poor)
n4 <- nrow(df_1)
n5 <- nrow(df_0)

m1 <- colMeans(df_grate[,1:11])
m2 <- colMeans(df_good[,1:11])
m3 <- colMeans(df_poor[,1:11])
m4 <- colMeans(df_1[,1:11])
m5 <- colMeans(df_0[,1:11])

s1 <- cov(df_grate[,1:11])
s2 <- cov(df_good[,1:11])
s3 <- cov(df_poor[,1:11])
s4 <- cov(df_1[,1:11])
s5 <- cov(df_0[,1:11])

sp <- ((n1-1)*s1+(n2-1)*s2+(n3-1)*s3)/(n1+n2+n3-3) #Spooled is HERE#
spi <- solve(sp)
spi

                     fixed.acidity volatile.acidity total.sulfur.dioxide
fixed.acidity          3110.857806        7.3185965           24.2149922
volatile.acidity          7.318596       28.0303581           -1.7038937
total.sulfur.dioxide     24.214992       -1.7038937            3.4452423
density              -55207.110776    -1399.9130816         -319.3604423
pH                     1638.546160        9.2556597           12.2910877
sulphates                29.824769        4.4528734           -0.3076922
residual.sugar           22.163691        0.2093982            0.1021050
chlorides               278.462380      -30.8393005            4.6354535
free.sulfur.dioxide      -1.073089        0.1036913           -0.1898212
alcohol                 -36.728423       -1.2446720            0.2364624
citric.acid            -126.463812       25.3255169           -3.5068357
                           density             pH      sulphates residual.sugar
fixed.acidity         -55207.11078   1638.5461603    29.82476937    22.16369060
volatile.acidity       -1399.91308      9.2556597     4.45287343     0.20939815
total.sulfur.dioxide    -319.36044     12.2910877    -0.30769222     0.10210499
density              1671992.60547 -29088.5007914 -1414.61288769  -705.46278416
pH                    -29088.50079   1673.6982443     7.71877174    12.09372726
sulphates              -1414.61289      7.7187717    12.88993094     0.69844840
residual.sugar          -705.46278     12.0937273     0.69844840     0.84517595
chlorides              -1677.20780    243.3817781   -26.14102361     0.18083640
free.sulfur.dioxide       26.39205     -0.7273122    -0.02061248    -0.02557939
alcohol                 1361.28259    -27.9828590    -1.62791964    -0.62976077
citric.acid             -983.33987     39.3121655    -0.44452752    -0.30768621
                         chlorides free.sulfur.dioxide        alcohol  citric.acid
fixed.acidity          278.4623803        -1.073088817  -36.728423413 -126.4638122
volatile.acidity       -30.8393005         0.103691339   -1.244671969   25.3255169
total.sulfur.dioxide     4.6354535        -0.189821166    0.236462372   -3.5068357
density              -1677.2077972        26.392049897 1361.282585942 -983.3398690
pH                     243.3817781        -0.727312174  -27.982858994   39.3121655
sulphates              -26.1410236        -0.020612480   -1.627919637   -0.4445275
residual.sugar           0.1808364        -0.025579391   -0.629760767   -0.3076862
chlorides              622.7774938        -0.168382670    4.727317172  -54.3304842
free.sulfur.dioxide     -0.1683827         0.020594188    0.005806883    0.1767275
alcohol                  4.7273172         0.005806883    2.435059048   -3.1048161
citric.acid            -54.3304842         0.176727524   -3.104816121   80.3449869

sp_2 <- ((n4-1)*s4+(n5-1)*s5)/(n4+n5-2) #Spooled is HERE#
spi_2 <- solve(sp_2)
spi_2

                     fixed.acidity volatile.acidity total.sulfur.dioxide
fixed.acidity          3108.141661        5.5979305           24.5672927
volatile.acidity          5.597930       27.4099149           -1.5802897
total.sulfur.dioxide     24.567293       -1.5802897            3.4230392
density              -55108.160031    -1351.4058851         -329.2106435
pH                     1631.987245        6.4562783           12.8469440
sulphates                30.309543        4.6280245           -0.3415658
residual.sugar           22.065924        0.1682393            0.1102374
chlorides               275.882362      -31.8772479            4.8374055
free.sulfur.dioxide      -1.075234        0.1032118           -0.1898337
alcohol                 -36.595349       -1.1877300            0.2253312
citric.acid            -127.562063       24.9645491           -3.4353962
                           density            pH      sulphates residual.sugar
fixed.acidity         -55108.16003   1631.987245    30.30954341    22.06592401
volatile.acidity       -1351.40589      6.456278     4.62802455     0.16823933
total.sulfur.dioxide    -329.21064     12.846944    -0.34156577     0.11023737
density              1669217.66640 -28889.593985 -1428.84070457  -702.70878282
pH                    -28889.59399   1662.413129     8.48144409    11.91976107
sulphates              -1428.84070      8.481444    12.85144297     0.71004077
residual.sugar          -702.70878     11.919761     0.71004077     0.84303333
chlorides              -1599.41289    239.055592   -25.88221530     0.11502708
free.sulfur.dioxide       26.45073     -0.730162    -0.02047829    -0.02563066
alcohol                 1357.66761    -27.746601    -1.64453352    -0.62641975
citric.acid             -954.78733     37.679928    -0.34300162    -0.33226680
                         chlorides free.sulfur.dioxide        alcohol  citric.acid
fixed.acidity          275.8823621        -1.075233605  -36.595348932 -127.5620633
volatile.acidity       -31.8772479         0.103211799   -1.187730031   24.9645491
total.sulfur.dioxide     4.8374055        -0.189833700    0.225331221   -3.4353962
density              -1599.4128909        26.450733628 1357.667611230 -954.7873300
pH                     239.0555920        -0.730161982  -27.746601290   37.6799281
sulphates              -25.8822153        -0.020478293   -1.644533522   -0.3430016
residual.sugar           0.1150271        -0.025630657   -0.626419754   -0.3322668
chlorides              621.5413949        -0.169357525    4.822436338  -54.9661725
free.sulfur.dioxide     -0.1693575         0.020606627    0.005859784    0.1765166
alcohol                  4.8224363         0.005859784    2.431362556   -3.0726683
citric.acid            -54.9661725         0.176516643   -3.072668332   80.1727517

box_m(df2[,1:11],df2[,"label"])

box_m(df2[,1:11],df2[,"y"])

Data is NO homogenious

Split data

df2$label <- with(df2, ifelse(label == 'great', 2, 
                         ifelse(label == 'good', 1, 0)))

set.seed(0)
sample <- sample.split(df2, SplitRatio = 0.7)

train  <- subset(df2, sample == TRUE)
test   <- subset(df2, sample == FALSE)

Logistic discrimination model

multi_model <- multinom(label ~ ., data = train[, 1:12])

# weights:  39 (24 variable)
initial  value 1216.163804 
iter  10 value 619.438991
iter  20 value 437.752258
iter  30 value 435.032107
iter  40 value 434.920188
iter  50 value 434.402810
iter  50 value 434.402806
final  value 434.402806 
converged

pred_multi <- predict(multi_model, newdata = test[, 1:12], type = "class")
correct_predictions <- sum(pred_multi == test$label)
correct_predictions

[1] 419

xtab <- table(pred_multi, test$label)
cm <- caret::confusionMatrix(xtab)
cm

Confusion Matrix and Statistics

          
pred_multi   0   1   2
         0   0   0   0
         1  22 398  47
         2   0   4  21

Overall Statistics
                                          
               Accuracy : 0.8516          
                 95% CI : (0.8171, 0.8819)
    No Information Rate : 0.8171          
    P-Value [Acc > NIR] : 0.02493         
                                          
                  Kappa : 0.3176          
                                          
 Mcnemar's Test P-Value : NA              

Statistics by Class:

                     Class: 0 Class: 1 Class: 2
Sensitivity           0.00000   0.9900  0.30882
Specificity           1.00000   0.2333  0.99057
Pos Pred Value            NaN   0.8522  0.84000
Neg Pred Value        0.95528   0.8400  0.89936
Prevalence            0.04472   0.8171  0.13821
Detection Rate        0.00000   0.8089  0.04268
Detection Prevalence  0.00000   0.9492  0.05081
Balanced Accuracy     0.50000   0.6117  0.64969

# Accuracy = TP / TOTAL
print(paste('Accuracy: ', (397 + 21) / 492 ))

[1] "Accuracy:  0.849593495934959"

# Recall = TP / (TP + FN)
Metrics::recall(pred_multi, test$label)

[1] 1.053533

 
# Precision = TP / (TP + FP)
Metrics::precision(pred_multi, test$label)

Warning: argument is not numeric or logical: returning NA

[1] NA

 
# F1 = 2 * (Precision * Recall) / (Precision + Recall)
Metrics::f1(pred_multi, test$label)

[1] 0.8

#𝐸(APER)
aer(test$label, pred_multi)

[1] 0.148374

model_3 <- lm(label ~ ., data=train[, 1:12], CV=TRUE)

Warning: In lm.fit(x, y, offset = offset, singular.ok = singular.ok, ...) :
 extra argument ‘CV’ will be disregarded

summary(model_3)

Call:
lm(formula = label ~ ., data = train[, 1:12], CV = TRUE)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.39692 -0.18171 -0.00273  0.11476  1.09962 

Coefficients:
                         Estimate   Std. Error t value     Pr(>|t|)    
(Intercept)           -0.56843323   1.06168373  -0.535      0.59248    
fixed.acidity          1.05831354   0.58307781   1.815      0.06979 .  
volatile.acidity      -0.29533790   0.05562280  -5.310 0.0000001329 ***
total.sulfur.dioxide  -0.00776280   0.01988918  -0.390      0.69639    
density              -34.46398517  13.56120834  -2.541      0.01118 *  
pH                    -0.36774762   0.42949206  -0.856      0.39205    
sulphates              0.21313571   0.03770351   5.653 0.0000000201 ***
residual.sugar         0.02182609   0.00998938   2.185      0.02911 *  
chlorides             -0.69182073   0.25312288  -2.733      0.00637 ** 
free.sulfur.dioxide   -0.00009178   0.00152391  -0.060      0.95199    
alcohol                0.07598378   0.01633404   4.652 0.0000036909 ***
citric.acid           -0.02654150   0.09526130  -0.279      0.78059    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3531 on 1095 degrees of freedom
Multiple R-squared:  0.2394,    Adjusted R-squared:  0.2317 
F-statistic: 31.32 on 11 and 1095 DF,  p-value: < 0.00000000000000022

plot(model_3, pch=df2$label)

pred_train_3 <- predict(model_3, train[, 1:12], type="response")
pred_test_3 <- predict(model_3, newdata = test[, 1:12], type="response")

train_TAB <- table(train$label, pred_train_3 > 0.95)
train_TAB

   
    FALSE TRUE
  0    27   14
  1   260  657
  2     1  148

test_TAB <- table(test$label, pred_test_3 > 0.95)
test_TAB

   
    FALSE TRUE
  0    10   12
  1   114  288
  2     1   67

model_binorm <- glm(y ~., data = train[, -12], family = binomial)
predictions <- predict(model_binorm, newdata = test[, -12], type = "response")
predicted_classes <- ifelse(predictions > 0.5, 0, 1)
mean(predicted_classes == test$y)

xtab <- table(predicted_classes, test$y)
cm <- caret::confusionMatrix(xtab)
cm

Confusion Matrix and Statistics

                 
predicted_classes   0   1
                0   5  22
                1 419  46
                                             
               Accuracy : 0.1037             
                 95% CI : (0.0782, 0.134)    
    No Information Rate : 0.8618             
    P-Value [Acc > NIR] : 1                  
                                             
                  Kappa : -0.0903            
                                             
 Mcnemar's Test P-Value : <0.0000000000000002
                                             
            Sensitivity : 0.01179            
            Specificity : 0.67647            
         Pos Pred Value : 0.18519            
         Neg Pred Value : 0.09892            
             Prevalence : 0.86179            
         Detection Rate : 0.01016            
   Detection Prevalence : 0.05488            
      Balanced Accuracy : 0.34413            
                                             
       'Positive' Class : 0                  
                                             

# Accuracy = TP / TOTAL
print(paste('Accuracy: ', (5 + 46) / 492 ))

[1] "Accuracy:  0.103658536585366"

# Recall = TP / (TP + FN)
Metrics::recall(predicted_classes, test$y)

[1] 0.09892473

 
# Precision = TP / (TP + FP)
Metrics::precision(predicted_classes, test$y)

[1] 0.6764706

 
# F1 = 2 * (Precision * Recall) / (Precision + Recall)
Metrics::f1(predicted_classes, test$y)

[1] 1

#𝐸(APER)
aer(test$y, predicted_classes)

[1] 0.8963415

#ROC -curve
roc_curve <- roc(ifelse(test$y == 0, 1, 0), ifelse(predicted_classes == 0, 1, 0))

Setting levels: control = 0, case = 1
Setting direction: controls < cases

# Plot ROC curve
plot(roc_curve, main = "ROC Curve", col = "blue")

fit <- vglm(label~., family=multinomial, data=train[, 1:12])
summary(fit)

Call:
vglm(formula = label ~ ., family = multinomial, data = train[, 
    1:12])

Coefficients: 
                         Estimate Std. Error z value       Pr(>|z|)    
(Intercept):1            3.618396  21.796586   0.166       0.868151    
(Intercept):2           16.366926  10.291605      NA             NA    
fixed.acidity:1         -3.402300  11.974217      NA             NA    
fixed.acidity:2         -8.647151   5.519332  -1.567       0.117184    
volatile.acidity:1       5.828700   1.041344   5.597 0.000000021773 ***
volatile.acidity:2       1.443152   0.575845   2.506       0.012206 *  
total.sulfur.dioxide:1  -0.214360   0.416160  -0.515       0.606491    
total.sulfur.dioxide:2   0.772187   0.231525   3.335       0.000852 ***
density:1              158.130232 274.549540   0.576       0.564641    
density:2              267.396317 131.762985   2.029       0.042420 *  
pH:1                    12.472557   8.730555   1.429       0.153116    
pH:2                     0.369114   4.304543   0.086       0.931665    
sulphates:1             -2.897919   0.739775  -3.917 0.000089547258 ***
sulphates:2             -2.572639   0.410467  -6.268 0.000000000367 ***
residual.sugar:1        -0.133223   0.195124  -0.683       0.494758    
residual.sugar:2        -0.207533   0.089742  -2.313       0.020747 *  
chlorides:1             10.217731   4.665817   2.190       0.028531 *  
chlorides:2              5.954070   3.462630   1.720       0.085519 .  
free.sulfur.dioxide:1   -0.004007   0.032871  -0.122       0.902977    
free.sulfur.dioxide:2   -0.022308   0.016551  -1.348       0.177720    
alcohol:1               -1.311135   0.346698  -3.782       0.000156 ***
alcohol:2               -0.639605   0.158989  -4.023 0.000057472394 ***
citric.acid:1            2.271508   1.769887   1.283       0.199345    
citric.acid:2           -0.254450   1.017662  -0.250       0.802561    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])

Residual deviance: 865.3682 on 2190 degrees of freedom

Log-likelihood: -432.6841 on 2190 degrees of freedom

Number of Fisher scoring iterations: 7 

Warning: Hauck-Donner effect detected in the following estimate(s):
'(Intercept):2', 'fixed.acidity:1', 'alcohol:1'


Reference group is level  3  of the response

probabilities <- predict(fit, test[,1:12], type="response")
predictions <- apply(probabilities, 1, which.max)
predictions[which(predictions==0)] <- test$label == 0
predictions[which(predictions==1)] <- test$label == 1
predictions[which(predictions==3)] <- test$label == 2

Warning: number of items to replace is not a multiple of replacement length

# summarize accuracy
xtab_v <- table(predictions, test$label)

cm <- caret::confusionMatrix(xtab_v)
cm

Confusion Matrix and Statistics

           
predictions   0   1   2
          0   0   5  20
          1   0   0   1
          2  22 397  47

Overall Statistics
                                             
               Accuracy : 0.0955             
                 95% CI : (0.071, 0.125)     
    No Information Rate : 0.8171             
    P-Value [Acc > NIR] : 1                  
                                             
                  Kappa : -0.0454            
                                             
 Mcnemar's Test P-Value : <0.0000000000000002

Statistics by Class:

                     Class: 0 Class: 1 Class: 2
Sensitivity           0.00000 0.000000  0.69118
Specificity           0.94681 0.988889  0.01179
Pos Pred Value        0.00000 0.000000  0.10086
Neg Pred Value        0.95289 0.181263  0.19231
Prevalence            0.04472 0.817073  0.13821
Detection Rate        0.00000 0.000000  0.09553
Detection Prevalence  0.05081 0.002033  0.94715
Balanced Accuracy     0.47340 0.494444  0.35148

# Accuracy = TP / TOTAL
print(paste('Accuracy: ', 47 / 492 ))

[1] "Accuracy:  0.0955284552845529"

# Recall = TP / (TP + FN)
Metrics::recall(predictions, test$labeæ)

Warning: argument is not numeric or logical: returning NA

[1] NA

 
# Precision = TP / (TP + FP)
Metrics::precision(predictions, test$label)

[1] 1.975124

 
# F1 = 2 * (Precision * Recall) / (Precision + Recall)
Metrics::f1(predictions, test$label)

[1] 1

#𝐸(APER)
aer(test$label, predictions)

[1] 0.9044715

PCA

num_data <- df2[, 1:11]
normalized <- scale(num_data)

#After that we create a correlation matrix
corr_matrix <- cor(normalized)
ggcorrplot(corr_matrix)

#Now we make a pca
data.pca <- princomp(corr_matrix)
summary(data.pca)

Importance of components:
                          Comp.1    Comp.2    Comp.3    Comp.4     Comp.5     Comp.6
Standard deviation     0.8915682 0.5626588 0.4458347 0.3287614 0.24273993 0.22153175
Proportion of Variance 0.5090581 0.2027442 0.1272934 0.0692181 0.03773467 0.03142897
Cumulative Proportion  0.5090581 0.7118023 0.8390956 0.9083137 0.94604842 0.97747739
                           Comp.7      Comp.8      Comp.9      Comp.10
Standard deviation     0.15155186 0.089660453 0.054877700 0.0339192702
Proportion of Variance 0.01470892 0.005148255 0.001928635 0.0007368026
Cumulative Proportion  0.99218631 0.997334563 0.999263197 1.0000000000
                                         Comp.11
Standard deviation     0.00000000533722824842028
Proportion of Variance 0.00000000000000001824273
Cumulative Proportion  1.00000000000000000000000

data.pca$loadings[, 1:2]

                          Comp.1      Comp.2
fixed.acidity         0.50837145  0.03432755
volatile.acidity     -0.27437604  0.34818335
total.sulfur.dioxide -0.08619193  0.42217508
density               0.36088060  0.34221286
pH                   -0.46124636 -0.11623776
sulphates             0.19289017 -0.22326201
residual.sugar        0.07676416  0.13981543
chlorides             0.16932322  0.15362777
free.sulfur.dioxide  -0.13509571  0.31081409
alcohol              -0.11698193 -0.58865323
citric.acid           0.46060706 -0.18061985

pca_1_2 <-data.pca$loadings[, 1:2]
pca_1_2<-as.matrix(pca_1_2)
numerical_data<-as.matrix(num_data)

#we multiply the numerical_data with our first and second pricipal components 

reduced_data<-numerical_data %*% pca_1_2
reduced_data<-as.data.frame(reduced_data)
reduced_data$predicted <- pred_multi <- predict(multi_model, 
                                                newdata = df2[, 1:12], type = "class")
reduced_data$true_class <- df2$label


plot1 <- ggplot(reduced_data, aes(x = Comp.1, y = Comp.2, colour = true_class)) +
  geom_point()

plot2 <- ggplot(reduced_data, aes(x = Comp.1, y = Comp.2, colour = predicted)) +
  geom_point()

plot1

plot2